A&A 388, 235-245 (2002)
DOI: 10.1051/0004-6361:20020473
J. A. Gil 1 - G. I. Melikidze 1,2 - D. Mitra 3
1 - Institute of Astronomy, University of Zielona Góra,
Lubuska 2, 65-265, Zielona Góra, Poland
2 - Center for Plasma
Astrophysics, Abastumani Astrophysical Observatory, Al.Kazbegi
ave. 2a, Tbilisi 380060, Georgia
3 - Max-Planck Institute for
Radioastronomy, Auf dem Hügel 69, 53121 Bonn, Germany
Received 27 November 2001 / Accepted 25 March 2002
Abstract
We propose a vacuum gap (VG) model which can be applied
uniformly for normal and high-magnetic-field pulsars. The model
requires a strong and non-dipolar surface magnetic field near the
pulsar polar cap. We assume that the actual surface magnetic field
in pulsars results from the superposition of a global
dipole field
and a crust-anchored small scale magnetic
anomaly
.
We provide a numerical formalism for
modelling such structures of the surface magnetic field and explore it
within the framework of the VG model, which requires strong surface
fields
G. Thus, in order to increase the
resultant surface field to values exceeding 1013 G, in low
magnetic field pulsars with
G it is required that
,
with the same polarities (orientations) of
and
.
However, if the polarities are opposite,
the resultant surface field can be lower than the dipolar surface
component inferred from the pulsar spin-down. We propose that
high-magnetic-field pulsars (HBPs) with the inferred global dipole
field
exceeding the so-called photon splitting threshold
G, can generate observable radio
emission "against the odds'', provided that the surface dipolar
magnetic field
is reduced below
by the magnetic
anomaly
of the right strength and polarity. We find that
effective reduction is possible if the values of
and
are of the same order of magnitude, which would be expected in
HBPs with
.
The proposed VG model of radio emission
from HBPs, in which pair production occurs directly above the polar
cap, is an alternative to the recently proposed lengthened
space-charge-limited-flow (SCLF) model, in which the pair formation
front is
located at relatively high altitudes, where the dipole field is
degraded below
.
Our model allows high
radio-loud
pulsars not only just above
but even above
G, which is the upper limit for HBPs within the
lengthened SCLF model.
Key words: pulsars: general
There are two important conclusions that can be drawn from the radio
emission properties of pulsars. Firstly the radio emission is thought
to arise at an altitude
from the center of the neutron star,
where
is of the order of several stellar radii,
R=106 cm (e.g. Kijak & Gil 1997, 1998, and references therein).
Secondly the regions from where the radio emission arises are consistent with
a purely dipolar magnetic field (Radhakrishnan & Cooke 1969).
However the structure of the magnetic field at the surface of the neutron
star is largely unknown. Strong non-dipolar surface magnetic fields
have long been thought to play an important role in the radio emission
of pulsars. For example, in order to sustain pair production in vacuum gaps,
the Ruderman & Sutherland (RS75) model implicitly assumed that the
radius of curvature of field lines above the polar cap should be about
106 cm, which is 100 times smaller than that expected from a global
dipolar magnetic field, thus indicating the presence of non-dipolar components.
It is believed that thermal X-rays from pulsars
are a good diagnostic tool to infer the structure of the surface magnetic field.
Soft X-ray observations of pulsars show non-uniform surface temperatures which
can be attributed to small scale magnetic anomalies on the pulsar polar cap
(e.g. Page & Sarmiento 1990; Bulik et al. 1992, 1995). Several similar arguments in favour of
the non-dipolar nature of the surface magnetic field can also be found in
Becker & Trümper (1997); Cheng et al. (1998); Rudak & Dyks (1999); Cheng & Zhang (1999); Thompson & Duncan (1995, 1996), Murakami et al. (1999), and Tauris & Konar (2001).
Also several theoretical studies concerning the formation and evolution of
non-dipolar magnetic fields in neutron stars are found in the literature
(e.g. Blandford et al. 1983; Krolik 1991; Ruderman 1991; Arons 1993; Chen & Ruderman 1993; Geppert & Urpin 1994; Mitra et al. 1999).
Woltjer (1964) proposed that the magnetic field in neutron stars results from the fossil field of the progenitor star which is amplified during the collapse stage and remains anchored in the superfluid core of the neutron star. It was also noted by several authors that shortly after or during the collapse of the neutron star magnetic fields can be generated in the outermost crust (e.g. by a mechanism like thermomagnetic instabilities; Blandford et al. 1983). Urpin et al. (1986) showed that in the crustal model it is possible to form small-scale surface field anomalies with typical sizes of the order of 100 meters. Further, Gil & Mitra (2001) demonstrated that such "sunspot''-like magnetic field structures on the polar cap surface help to sustain VG-driven radio emission of pulsars. In this paper we consider the scenario where the magnetic field on the neutron stars' surface is non-dipolar in nature which is due to superposition of the fossil field in the core and the crustal field structures. The crust gives rise to small-scale anomalies which can be modelled by a number of crust-anchored dipoles oriented in different directions (e.g. Blandford et al. 1983; Arons 1993). The superposition of global dipole and local anomaly is illustrated in Fig. 1, where for clarity of presentation only one local, crust-associated dipole is marked.
Formation of dense electron-positron pair plasma is essential for
pulsar radiation, especially (but not only) at radio wavelengths.
A purely quantum process for magnetic pair production
is commonly invoked as a source of this
plasma (e.g. Sturrock 1971; Ruderman & Sutherland 1975). For superstrong magnetic
fields close to the so-called quantum field
G, the process of free e-e+ pair production can be
dominated by the phenomenon of photon splitting
(Adler et al. 1970; Bialynicka & Bialynicki et al. 1970; Baring & Harding 1998) and/or bound positronium formation
(Usov & Melrose 1995, 1996). While the latter process can reduce the number
of free pairs at magnetic fields
(e.g. Baring & Harding 2001), the former can suppress the
magnetic pair production at
G entirely, provided that
photons which are polarized both parallel and perpendicular to the local
magnetic field direction can split (e.g. Baring 2001; Baring & Harding 2001). This
assumption will be implicitly made throughout this paper. Under
these circumstances one can roughly define a photon splitting
critical line
and expect that there should be no
radio pulsar above this line on the
diagram, where
G is the dipole surface
magnetic field estimated at the pole from the pulsar period Pand its derivative
(Shapiro & Teukolsky 1983; Usov & Melrose 1996). This death-line is
more illustrative than quantitative. In fact, a number of specific
model-dependent death-lines separating radio-loud from radio-quiet
pulsars are available in the literature
(Baring & Harding 1998, 2001; Zhang & Harding 2000a, 2001). All these slightly period-dependent
death-lines cluster around
on the
diagram, and
hence the quantum critical field is conventionally treated as a threshold
magnetic field above which pulsar radio emission ceases. In this
paper we also use this terminology, bearing in mind that the
photon splitting threshold realistically means a narrow range of
magnetic fields around the critical quantum field
G, certainly above 1013 G
(see review by Baring 2001). For convenience, in all numerical examples presented
in Figs. 2-6 and subsequent discussions we assume that the threshold
magnetic field
.
In order to produce the necessary dense electron-positron plasma,
a high-voltage accelerating region has to exist near the polar cap
of pulsars. Two models of such acceleration regions are available
in the literature, namely: the
stationary space charge limited flow (SCLF) model
(Sharleman et al. 1978; Arons & Sharleman 1979; Arons 1981) in which charged particles flow freely from
the polar cap, and the highly non-stationary vacuum gap (VG) models
(Ruderman & Sutherland 1975; Cheng & Ruderman 1977, 1980; Gil & Mitra 2001) in which the free outflow of charged
particles from the polar cap surface is strongly impeded. In the
VG models the charged particles accelerate within a height scale
of about the polar cap radius (i.e. 104 cm), due to a high potential
drop across the gap, while in the SCLF models particles accelerate
within a height scale of a stellar radius
106 cm, due to
the potential drop resulting from the curvature of field lines
and/or the inertia of outstreaming particles. In both models the free
e-e+ pairs are created if the kinematic threshold
is reached or
exceeded and the local magnetic field is lower than the photon
splitting threshold
,
where
is the photon energy and
is the propagation angle with respect to the
direction of the local magnetic field.
Recent discovery of high-magnetic-field pulsars (HBPs) however has
challenged the existing pair creation theories. Few HBPs are seen to have
the inferred surface dipolar fields above the photon splitting
level: PSRs J1119-6127, J1814-1744 and J1726-3530 (Table 1).
Yet another strong-field neutron star PSR J1846-0258
with
G was discovered (Gotthelf et al. 2000),
which seems to be radio-quiet (Kaspi et al. 1996), although its X-ray
emission is apparently driven by dense
pair plasma
(e.g. Cordes 2001). Bearing in mind that the
actual threshold due to photon splitting and/or bound positronium
formation can be well below the critical field
G, all high-magnetic-field radio pulsars
with
G pose a challenge. To evade the photon
splitting problem for these pulsars
Zhang & Harding (Zhang & Harding 2000a, ZH00 hereafter) proposed "a unified picture for HBPs and
magnetars''. They argued that radio-quiet magnetars cannot have
active inner accelerators (thus no
pair production),
while the HBPs can, with a difference attributed to the relative
orientations of rotation and magnetic axes (neutron stars can be
either parallel rotators (PRs) with
or
antiparallel rotators (APRs) with
,
where
is the pulsar spin axis and
is the
magnetic field at the pole). If the photon splitting suppresses
completely the pair production at the polar cap surface, then the
VG inner accelerator cannot form, since the high potential drop
cannot be screened at the top of the acceleration region. Hence,
ZH00 argued that in the high magnetic field regime
the
pair production process is possible only if the SCLF accelerator
forms. In fact, such SCLF accelerators are typically quite long
and their pair formation front (PFF) can occur at high altitudes
r, where the dipolar magnetic field
has
degraded below
.
Furthermore, ZH00
demonstrated that such a lengthened SCLF accelerator in a magnetar
environment can form only for PRs and not for APRs. Consequently
they concluded that the radio-loud HBPs are PRs with developed
lengthened SCLF accelerator, while the radio-quiet magnetars (AXPs
and SGRs) are APRs. It is worth emphasizing here that ZH00
developed their model under the assumption that the magnetic field
at the surface of HBPs is purely dipolar.
source | P (s) | ![]() |
Bp (G) |
PSR J1814-1744 | 3.98 |
![]() |
![]() |
PSR J1119-6127 | 0.41 |
![]() |
![]() |
PSR J1726-3530 | 1.11 |
![]() |
![]() |
In this paper we propose an alternative model for radio-loud HBPs
based on a highly non-dipolar surface magnetic field, in which the
photon splitting within the VG inner acceleration region does not
operate even if the dipole magnetic field exceeds the critical
value
at the polar cap. Thus our model requires that HBPs
are APRs, which is a consequence of the VG scenario
(e.g. Ruderman & Sutherland 1975; Gil & Mitra 2001). This model is a follow-up of
Gil & Mitra (2001), who argued that the VG can form if the actual
surface magnetic field is about 1013 Gauss. In other words,
they assumed that all VG-driven radio pulsars require strong
highly non-dipolar surface magnetic fields, with strength
being more or
less independent of the value of the global dipole field inferred
from the magnetic braking law. Thus, if
G then
and if
G then
.
However, for radio pulsars to operate, in any case
.
We argue that such
strong surface field anomalies can increase the dipolar field
in normal pulsars to values exceeding 1013 G
(required by the conditions for VG formation - see Gil & Mitra 2001)
if the global and local surface fields have the same polarities, or
reduce the very high dipolar field
in HBPs, if
both these components are of comparable values and have opposite
polarities.
![]() |
Figure 1:
Superposition of
the star-centered global magnetic dipole "d'' and crust-anchored
local dipole "m'' placed at
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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We model the actual surface magnetic field by superposition of
the star-centered global dipole
and a crust-anchored
dipole moment
,
whose influence results in small-scale
deviations of the surface magnetic field from the global dipole, as
presented in Fig. 1. The technical details of the calculations of
the resultant surface magnetic field are presented in the Appendix.
Here we discuss the main results and their consequences. For
simplicity, in this paper we mostly consider an axially symmetric
case in which both
and
are directed
along the z-axis (parallel or antiparallel), thus
(see Fig. 1). Also, for
convenience "
'' is expressed in units of
.
We use normalized units in which d=P=R=1 and
.
(see
caption of Fig. 2 for the normalization convention). All
calculations are carried out in three-dimensions, although, for
clarity of graphic presentation, in the figures we present only
two-dimensional cuts of the open field line regions.
![]() |
Figure 2:
Structure of
the surface magnetic field for a superposition of the global star-centered
dipolar moment ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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As mentioned above, the formation of the VG inner accelerator requires
a very high magnetic field
G on the surface of
the polar cap (Usov & Melrose 1995, 1996; Gil & Mitra 2001). This can be achieved not
only in pulsars with high dipolar field
G. In
fact, some of the low-field pulsars with
G can
have a surface field
G if
(thus
d). We discuss such normal, low-field
pulsars later in this paper. Presently let us consider the HBP
with a dipolar surface field at the pole
G exceeding the photon splitting
limit
.
If all photon splitting modes operate,
such a pulsar should be radio-quiet. Alternatively these pulsars
could be radio-loud if the effective surface field is reduced
below
.
Such a scenario can be achieved if the polarities of
magnetic moments
and
are opposite, that is
and
are antiparallel. Figure 2 presents
a case with
and
.
The
actual surface values of
as
well as radial components of
and
are presented in the lower panel
of Fig. 2 (note that all radial components are positive
and that the total
is almost equal to
in this case). At
the pole (radius r=R and polar angle
)
the ratio
and thus
.
As one can see from this figure,
all surface field lines between
and
are
open, but the ratio
increases towards the polar cap
edge, reaching a value of about 0.5 in the region between polar
angles
and
.
The ratio
is also
about 0.5 in this region. Thus, the global dipolar field (
in our units) is effectively reduced by between 2 and 5 times in
different parts of the polar cap (defined as the surface area from
which the open magnetic field lines emanate). This means that the
ratio
ranges from 0.5 to 0.8 across the polar cap. The
actual polar cap is broader than the canonical dipolar polar cap
(two dashed vertical lines correspond to the last open dipolar field
lines emanating at the polar angles
radians
for a typical period P=1 s). The ratio of actual to dipolar polar
cap radii is
in this case. Thus, using
the argument of magnetic flux conservation of the open field
lines, one can say that an effective surface magnetic field of the
polar cap is about 2.8 times lower than the dipolar surface field
measured from the values of P and
.
If the estimated
dipole field
G (like in the case of PSR
J1814-1744, Table 1) then the actual surface field at the pole
is only
G, well below the photon
splitting death line
G. Such a pulsar can
be radio-loud without invoking the lengthened SCLF accelerator
proposed by ZH00. As shown by Gil & Mitra (2001), in such a strong surface
magnetic field the vacuum gap accelerator can form, which implies
low altitude coherent radio emission
(Melikidze et al. 2000, see Sect. 3 in this paper) at altitudes
(for a typical
pulsar with P=1 s) in agreement with observational constraints
on radio emission altitudes (Cordes 1978, 1992; Kijak & Gil 1997, 1998; Kijak 2001).
![]() |
Figure 3:
As in
Fig. 2 but for
![]() |
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Figure 3 presents another case of opposite polarities
,
with magnitude of
being two
times stronger than in the previous case (Fig. 2). Again
for
,
at
the pole (r=R and
)
and
.
The
negative sign of the ratio
means that the surface
magnetic field
is directed opposite to
near the pole, that is the circumpolar field lines with polar
angles
(where
)
are closed. The last open surface field lines (solid)
emanating at polar angles
(where
)
reconnect with the last open dipolar field
lines (dashed) at altitudes
(thus about 2 km above
the surface). The actual polar cap, which is the surface through
which the open magnetic field lines emanate, has the shape of a ring
located outside the
circle of the dipolar polar cap with angular radius
(or diameter
cm).
Again, the magnetic flux conservation argument leads to
,
thus
is about
within the ring of the open field
lines (note that
in this region). The actual values
of the surface magnetic field (radial
and total
)
are shown as solid lines in
the lower panel of Fig. 3, in comparison with radial
components of the dipolar field
(dashed horizontal line). As one
can see,
and
is negative
for
.
If the dipolar surface component of the pulsar
magnetic field is
G (like PSR
J1726-3530 given in Table 1), then the actual surface magnetic field
G, is below the photon splitting threshold.
![]() |
Figure 4:
As in
Fig. 3 but for
![]() |
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Figure 4 presents a case with
in which both magnetic moments have the same polarity.
Obviously, in such a case the surface magnetic field will be
stronger as compared with pure dipole (m=0). The flux conservation
argument gives a surface magnetic field
(the ratio
). Thus the actual
surface field is about 3 times stronger than the inferred dipolar
field
G.
Such cases of
increasing an effective magnetic field can be important in normal
pulsars with low dipolar field
G (Gil & Mitra 2001).
![]() |
Figure 5:
As in
Fig. 4 but for
![]() |
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It is then interesting to examine how different polarities of
and
would influence normal pulsars with
.
If
thus
then of course
can be slightly lower than
,
as in the
case of HBPs (Fig. 2). In such a case, however, the VG
cannot form. In fact, as argued by Gil & Mitra (2001), the formation of
the VG requires that
is close to 1013 G or even above, thus
is required in normal pulsars (see also
Gil et al. 2001). Figure 5
illustrates a case of high surface magnetic field with
,
in which the VG can apparently form. As one can see from this
figure, the values of
at the ring-shaped polar cap are close
to dipolar values
.
One can show
that this is a general situation, that is,
no matter
by how much
exceeds
at the pole. This follows from the
fact that the angular location
of the polar cap ring
increases with increasing ratio
.
For example, in the case
presented in Fig. 5
and
the last open field lines emanate at polar angles
radians, or at polar cap radii
cm (for P=1 s). Thus, the narrow polar
cap ring is located far from the local dipole
,
whose
influence is weak at this distance. The circumpolar field lines
between polar angles -0.053 to +0.053 are closed.
Thus, we conclude that the actual pulsar surface magnetic field
can significantly differ (say by an order of magnitude) from
the inferred dipolar field
only in the case when the
polarities of the global
and local
dipole
(Fig. 1) are the same, as illustrated in Fig. 4.
If this is the case, then
can largely exceed
,
which
seems to be important from the viewpoint of vacuum gap formation,
which requires
G (see Gil & Mitra 2001). Therefore, in
normal VG-driven radio pulsars the polar cap should be circular,
or at least filled - if the axial symmetry does not hold. A
ring-shaped polar cap can occur only in normal pulsars with
and in radio-loud HBPs with
.
In the accompanying paper Gil et al. (2002) explored consequences of
the vacuum gap model interpretation for drifting subpulses observed
in PSR B0943+10, in which 20 sparks move circumferentially around the
perimeter of the polar cap, each completing one circulation in 37
pulsar periods (Deshpande & Rankin 1999, 2001). Gil et al. (2002) considered both
the curvature radiation (CR) and resonant inverse Compton
radiation (ICS) seed photons as sources of electron-positron
pairs and determined the parameter space for the surface magnetic
field structure in each case. For the CR-VG the surface magnetic
field strength
G and the radius of
curvature of surface field lines
,
while for the resonant ICS-VG
G and
(of
course, in both caseS
G). The
CR-VG with such a curved surface magnetic field does not seem likely
(although it cannot be excluded), while the ICS-VG gap supported
by the magnetic field structure determined by the parameter space
specified above guarantees a system of 20 sparks circulating
around the perimeter of the polar cap by means of the
drift in about 37 pulsar periods.
Further, Gil et al. (2002) modelled the magnetic field structure
determined by the ICS-VG parameter space (specified above), using
the numerical formalism developed in this paper. Since
G in this case, to obtain
G one
needs
and the same
polarity of both components. Following the symmetry suggested by
the observed patterns of drifting subpulses in PSR B0943+10, the
local dipole axis was placed at the polar cap center. A number of
model solutions corresponding to
and
d and satisfying the ICS-VG parameter space, was
then obtained. As a result of this specific modelling
Gil et al. (2002) obtained a number of interesting and important
conclusions: (i) The conditions for the formation of the ICS-VG
are satisfied only at the peripheral ring-like region of the polar
cap, which can just accommodate a system of 20 sparks performing
drift. (ii) The surface magnetic field lines
within the actual gap are converging, which stabilizes the
drifting sparks by preventing them
from rushing towards the pole, as opposed to the case of a diverging
dipolar field (e.g. Fillipenko & Radhakrishnan 1982). (iii) No model solutions with
G and
cm, could be obtained which corresponding to the
CR-VG parameter space, which in turn favors the ICS-VG in PSR B0943+10.
We argue in this paper that the presence of a strong
non-dipolar magnetic field on the neutron star surface can help to
understand the recently discovered radio pulsars with dipolar
magnetic field above the photon splitting threshold, as well as to
understand the long-standing problems of vacuum gap formation and
drifting subpulse phenomenon. We model the actual surface magnetic
field as a superposition of the global star-centered (large-scale)
dipole and local crust-anchored (small-scale) dipoles
,
where
is the local dipole nearest to the polar cap
centre (Fig. 1). Such a model is quite general, as it describes the
magnetic field structure even if the star-centered fossil dipole field is
negliglible at the star surface. In such a case the global surface dipole
field
(inferred from P and
measurements)
is a superposition of all crust-anchored dipoles calculated at a far
distance and projected down to the polar cap surface
according to the dipolar law.
We propose a model for radio-loud HBPs with high inferred dipolar
magnetic field
G, even exceeding the critical value
G. Given the difficulty that in a
strong magnetic field the magnetic pair creation process is
largely suppressed, the puzzling issue remains how these HBPs
produce their
pair plasma necessary for the generation of the
observable radio emission. Zhang & Harding (2000a) proposed a "lengthened
version'' of the stationary SCLF model of inner accelerator
(e.g. Arons & Sharleman 1979), in which the pair formation front occurs at
altitudes r high enough above the polar cap that
degrades below
,
thus evading the photon
splitting threshold. Our VG model is an alternative to the
lengthened SCLF model, with pair creation occurring right at the
polar cap surface, even if the magnetic field exceeds
.
We
have assumed that the open surface magnetic field lines result in
an actual pulsar from superposition of the star-centered global
dipole moment and a crust-anchored local dipole moment. We argued
that if the polarities of these two components are opposite, and
their values are comparable, then the actual value of the surface
magnetic field
can be lower than the critical field
,
even if the global dipole field
exceeds the
critical value. Thus, the creation of electron-positron plasma is
possible at least over a part of the polar cap and these high
magnetic field neutron stars can be radio-loud (HBPs). In fact,
one should expect that in HBPs, in which by definition
G, the ratio
should be
of the order of unity, since
and
G.
Within our simple model of a non-dipolar surface magnetic field
one should expect that both cases
and
will occur with approximately
equal probability. However, from the viewpoint of observable radio
emission only the latter case is interesting in HBPs with
.
In fact, when
,
then
the surface magnetic field
(Fig. 4)
and the photon splitting level is highly exceeded. For
we have two possibilities: (i) if
thus
at the pole
then the polar cap (locus of the open field lines) is
circular (Fig. 2); (ii) if
thus
then part of the circumpolar field lines are closed and
the actual polar cap has the shape of ring (Fig. 3). In
both cases (i) and (ii), the actual surface magnetic field
at the polar cap (or at least part of it) can be lower than
,
even if
exceeds
.
The values of
and
should be comparable to make reduction of
The strong surface field
below
possible. In our
illustrative examples presented in Figs. 2 and 3 (corresponding to
the same pulsar with P=1 s and
G) we used ratios
ranging
from 0.5 to 1.6. These values could be slightly different, say by
a factor of a few, thus we can say that the ratio
should
be of the order of unity. If
,
then the reduction of
the surface dipole field is not effective (see example presented and
discussed in Fig. 5). On the other hand, the case with
is not interesting, as it represents a weak surface magnetic
field anomaly. Thus, among a putative population of neutron stars
with
,
only those with a ratio
of the order of unity, and with
magnetic moment
and
(Fig. 1) antiparallel at
the polar cap surface, that is
,
can be
detected as HBPs. Other neutron stars from this population of high
magnetic dipole field objects should be radio-quiet. This probably
explains why there are so few HBPs detected.
Within the lengthened SCLF model there is an upper limit around
G for radio-loud HBPs (ZH00, ZH01, Zhang 2001).
As ZH00 argued, detecting a pulsar above this limit would strongly
imply that only one mode of photon splitting occurs. Without the
alternative model of HBPs proposed in this paper, such a detection
would really be of great importance for the fundamental physics of
the photon splitting phenomenon. In our VG-based model there is no
natural upper limit for the radio-loud HBPs. However, it is known
that due to the magnetic pressure the neutron star surface would
tend to "crack'', which should occur at magnetic field strengths
approaching 1015 G (Thompson & Duncan 1995). It is unclear how the radio
emission would be affected by such a cracking process.
![]() |
Figure 6:
As in
Fig. 2 but for
![]() |
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To illustrate the above argument, let us consider Fig. 6
which presents yet another case of opposite polarities
.
With
this
gives
and
at the pole (r=R,
). The dashed horizontal line at B=0.2 in the lower
panel corresponds to the surface magnetic field
which is 10
times weaker than the global dipole component
(not shown
in the figure). Thus if, for example,
G (well
above the lengthened SCLF limit
G; such
a pulsar was not observed so far), then the actual surface field
is well below
G, at least in
the inner part of the polar cap between
rad.
This "pair-forming effective'' polar cap is about 2.5 times
smaller than the canonical polar cap with radius
rad, and about 7 times smaller than the entire
polar cap with radius
rad. Near the last open
field lines at polar angles
the actual surface magnetic field
is only about 2 times
lower than
,
while in a narrow circumpolar area with
the surface field region
can even be
more than 10 times weaker than
.
Thus, within our model one
can expect a radio-loud HBP with
even exceeding
G. However, their radio beams should be much
narrower than those expected in normal pulsars, at least a few to
several times less than
degrees
(where
is the radio emission altitude; Kijak & Gil 1997, 1998). This would make such sources difficult to
detect.
![]() |
Figure 7:
As in
Fig. 6 but with the local dipole shifted off center by
![]() |
Open with DEXTER |
The dotted horizontal line at B=0.05 in Fig. 6
corresponds to
G for adopted
G. This value of the surface magnetic field is believed
to be about the lower limit for VG formation
(see Gil & Mitra 2001; Gil et al. 2002). Thus, the shadowed area in
Fig. 6 represent a narrow hollow cone above which the VG-driven
radio emission cannot occur. A similar hollow cone is marked
in Fig. 7, which presents a case similar to that
illustrated in Fig. 6, except that the local dipole is shifted
off center by
radians (corresponding to about
0.2 of the actual polar cap radius). The dashed horizontal line at
B=0.4 corresponds to
G and the dotted
horizontal line at B=0.1 corresponds to
G, both
calculated for adopted
G. The polar angles
and
correspond to
and
in Fig. 6, respectively. Figure 7
demonstrates that the conclusions of our paper do not depend on where
the local dipole is placed.
The above arguments strengthen the possibility that some magnetars
can also emit observable radio emission (Camilo et al. 2000; Zhang & Harding 2000b). It
is therefore interesting to comment on the apparent proximity of
HBP PSRJ 1814-1744 (with
G) and AXP 1E
2259+586 (with
G) in the
diagram. In both these cases the inferred surface magnetic field
well exceeds the critical value
.
Within our model, the
former object can be radio-loud if the strong local dipole and the global dipole
have opposite polarities. The radio
quiescence of AXP 1E 2259+586 can be naturally explained if the
local dipole is not able to decrease the inferred dipole magnetic
field below the photon-splitting death-line. Thus, either the
polarities are the same or they are opposite but the local dipole
is not strong enough to reduce the dipole surface field below
.
![]() |
Figure 8:
Radius of
curvature
![]() ![]() |
Open with DEXTER |
In Fig. 8 we show the radii of curvature of actual
surface field lines compared with those of purely dipolar field
(line 1) as a function of normalized altitude
above the polar cap. Within the polar gap at z<1.01 (within
about 100 meters from the surface) the curvature radii for all
cases presented in Figs. 2-6 have values of the order of a few
hundred meters (see Urpin et al. 1986), suitable for
curvature-radiation-driven magnetic pair production
(
cm, where
is the curvature of the field
lines).
All model calculations performed in this paper correspond to the axisymmetric case in which one local dipole is placed at the polar cap center (except the case presented in Fig. 7). In a forthcoming paper we will consider a general, non-axisymmetric case, including more local dipoles, each with different orientation with respect to the global dipole. Although this generalization will give a more realistic picture of the actual surface magnetic field, it will not change our conclusions obtained in this paper.
It should be finally emphasized that although the lengthened SCLF
model for HBPs (ZH00) can solve the problem of pair creation in
pulsars with surface dipole field exceeding the photon splitting
threshold, it does not automatically warrant generation of the
coherent radio emission of such HBPs. The problem is that unlike
in the non-stationary VG model, where the low altitude radio
emission can be generated by means of two-stream instabilities
(Asseo & Melikidze 1998; Melikidze et al. 2000), the stationary SCLF inner accelerator is
associated with the high-altitude relativistic maser radiation
(e.g. Kazbegi et al. 1991, 1992; Kazbegi et al. 1996). This radiation requires
relatively low Lorentz factors
of a dense
secondary plasma (e.g. Machabeli & Usov 1989). It is not clear if such a
plasma can be produced within the lengthened SCLF accelerator with
delayed pair formation taking place in a purely dipolar magnetic
field, either by curvature radiation or by inverse Compton
scattering (e.g. Zhang & Harding 2000b) processes. Moreover, the
relativistic maser coherent radio emission requires a relatively
weak magnetic field in the generation region. With the surface
dipole field
G, such a low field may not exist at
reasonable altitudes (about 50% of the light cylinder radius
)
required by the physics of corresponding
instabilities (Kazbegi et al. 1991, 1992; Kazbegi et al. 1996). Thus, if one assumes
that the radio-loud HBPs are driven by the SCLF lengthened
accelerator as proposed by ZH00, they might not be able to
generate observable coherent radio emission. This contradiction
seems to be a challenge for the lengthened SCLF scenario for HBPs.
In our VG-based model the low-altitude
radio emission
of HBPs is driven by just the same mechanism as the one most
probably operating in typical radio pulsars
(e.g. soliton curvature radiation proposed recently by Melikidze et al. 2000). In fact, the
HBPs show apparently normal radio emission, with all properties
typical for characteristic pulsar radiation (Camilo et al. 2000).
In order to model the actual surface magnetic field by
superposition of the star-centered global dipole
and
the crust-anchored dipole moment
,
let us consider the
general geometrical situation presented in Fig. 1. The
resultant surface magnetic field is
![]() |
(A.1) |
The equations of the line of force of the vector field
have a well known form in spherical
co-ordinates
![]() |
![]() |
![]() |
= | ![]() |
|
![]() |
= | ![]() |
|
![]() |
= | ![]() |
![]() |
= | ![]() |
|
![]() |
= | ![]() |
|
![]() |
= | ![]() |
![]() |
(A.6) |
![]() |
(A.7) |
The curvature
of the field lines (where
is
the radius of curvature presented for various cases in Figs. 2-8)
is calculated as
![]() |
(A.9) |
J1 | = | ![]() |
|
J3 | = | Z2S1-Z1S2 | |
X1 | = | ![]() |
|
Y1 | = | ![]() |
|
Z1 | = | ![]() |
|
X2 | = | ![]() |
|
![]() |
|||
Y2 | = | ![]() |
|
![]() |
|||
Z2 | = | ![]() |
|
S1 | = | ![]() |
|
S2 | = | ![]() |
|
![]() |
![]() |
||
![]() |
![]() |
![]() |
(A.11) |
Acknowledgements
This paper is supported in part by the KBN Grant 2 P03D 008 19 of the Polish State Committee for Scientific Research. We are grateful to Dr. U. Geppert for very helpful discussions. We also thank E. Gil and G. Melikidze Jr. for technical help. DM would like to thank the Institute of Astronomy, University of Zielona Góra, for support and hospitality during his visit to the institute, where this and the accompanying paper were started.